Efficient numerical algorithms of time fractional telegraph‐type equations involving Hadamard derivatives

Abstract

In this paper, we study the finite difference/iterative method for the fractional telegraph equation with Hadamard derivatives. The model is first transformed into an equivalent fractional integro‐differential equation, then the technic of exponential type meshes is adopted. The resulting discrete coefficients of Hadamard fractional integral own several graceful properties, which are crucial in numerical analysis. To conquer the difficulty caused by weak regularity, we also propose iterative schemes. The error estimate is equal to O()false(logtfalse)n+γ,0.1em1<t<e$$ O\left({\left(\log t\right)}&amp;#x0005E;{n&amp;#x0002B;\gamma}\right),1&lt;t&lt;e $$, where γ$$ \gamma $$ is the order of fractional derivative and n$$ n $$ is the number of iterations. All the theoretical results are confirmed by numerical experiments.